Dedicated to the application of mathematical modeling and fieldwork to the study of the history of cartography...and perhaps a few philosophical reflections on the above...

Author Climbing in the Queyras, Summer 2013

Tuesday, February 07, 2012

Modeling Roman Land Use and Environment:Epigraphy, Servitudes, and Game Theory

...we have all too often lacked, or failed to consider, conceptual frameworks of theory in which to examine Man's relationship to his environment, the manner in which he weighs the alternatives presented, and the rationality of his choices once they have been made....---Peter Gould

In the study of Roman agricultural patterns it is important to have a conceptual framework in which to place the fragmentary information and evidence that is available from epigraphy, Roman law, and landscape archaeology. For the past few months I have been experimenting with Game Theoretical Models and the concept of Nash Equilibrium trying to see what type of land use models would arise.

The basis of game theory was first laid down in the late 1940's by the mathematician John vonNeumann and the economist OskarMogenstern in their now classic book

the Theory of Games and Economic Behavior. In the book vonNeumann gives the proof of the Minimax Theorem, which is central to game theoretic reasoning and that he first approached in 1928. In the 1944 book, vonNeumann placed the theorem within the context of linear inequalities and the theory of convexity, which was later updated with more formal proofs of equilibrium states by John Forbes Nash.

My current work on modeling land use and some of the environmental decisions made by Roman farmers takes its real start however, from a conversation that I had with Waldo Tobler, Emeritus Professor of Geography at California, Santa Barbara, about 8 years ago. I had just read Peter Gould’s paper on African farmers in General Systems Theory, a paper that would later lead me to his seminal work, Man against the Environment. I knew that Tobler was close to Gould and that he was also playing around with some game theory during these years, and so I asked Tobler about the paper. What was most impressive to me in all this was not really Gould’s mathematics, but rather his vision of what game theory might be able to do in geographical sciences, that even simple matrix games had a spatial component that few geographers had thought to utilize.

One of the things that Gould wrote and that struck me as profound was that , “we have all too often lacked, or failed to consider, conceptual frameworks of theory in which to examine Man's relationship to his environment, the manner in which he weighs the alternatives presented, and the rationality of his choices once they have been made.” The rationality part instantly jumped out at me. As you may or may not know, the idea of rationality is an area of hot debate when it comes to questions of the Roman economy. There are many scholars, especially after Finley’s seminal book called The Ancient Economy, who believe that to consider Roman farmers and landowners as ‘rational’, in the sense of their maximizing the yield from their farms and thinking about market forces, is to project too much of a modern conception of a market economy onto the past. More recently however, some scholars like Dennis Kehoe, Cynthia Jordan Bannon and D. W. Rathbone, using legal inscriptions and the everyday account books of farms that survive as papyrus fragments, have started to use economic models and things like the theory of the commons to talk about Roman markets and agricultural estate management. Each of them in their own way incorporates many of the terms and categories of game and decision theory in their analysis. Perhaps the best book that accepts and summarizes the presence of ‘rational’ actors in the Roman economy is a book by Paul Erdkamp, entitled, The Grain Market in the Roman Empire: a social, political and economic study. Erdkamp puts forward many different models in the book, and summarizes the economic theory in his historical examinations and reconstructions. His is the sort of book that makes you anxious when you read it, as it gives you a good idea of how much you do not know and how long it takes to make any real progress in this area.

My own models are simply extensions of this kind of work. One group of Gould’s papers, from which my research certainly takes its inspiration, was written by him in the 1960's. His papers, "Wheat on Kilimanjaro: the perception of choice in game and learning model frameworks," and "Man against His Environment: a game theoretic framework", were among the first attempts to use the concepts of game theory and Nash equilibrium to look into agricultural land use. These papers, and a few others, were also discussed in an early review article on these methods written by David Harvey, "Theoretical Concepts and the Analysis of Agricultural Land-Use Patterns in Geography." It is in fact from Harvey’s book, Models in Geography that my concept of geographic model derives.

Harvey asserts, in his review article on agricultural land use, that at the time he was writing, many geographers tended to ignore theoretical breakthroughs from other disciplines, mainly on the "grounds that they proved too abstract to help in the search for unique causes of specific events." To counter this he quotes from William Bunge, whose book Theoretical Geography transformed geography and opened up an analytical window for the field, suggesting a more theoretical and inherently mathematical approach to the study of geographical and spatial distributions. To me Bunge’s book is the most important work of geography in the 20th century and I still mine it for inspiration.

Most of Harvey's paper is dedicated to outlining the requirements for a set of theoretical and conceptual elements to constitute a model in geography. A model, according to Harvey, requires a set of relationships to be established that somehow link the input, status and output variables in a specific way. This linkage must quantify the model mathematically in order for it to be tested. For Harvey, the relationships of the variables in the model can be of three distinctive types:

1. Deterministic relationships which specify cause and effect sequences.2. Probabilistic relationships which specify the likelihood of a particular cause leading to a particular effect.3. Functional relationships which specify how two variables are related or correlated without necessarily having any causal connection at all.

For agricultural models Harvey makes a distinction between two types of frameworks, one in which the underlying structure is normative and therefore, describes what ought to be under certain assumptions. The second, is descriptive, and describes what it is that exists. These distinctions are extremely important when we try to interpret game theoretical models, especially in something as difficult to conceptualize as the Roman economy.

In his early research Gould, using a normative game theoretic model, studied a group of African farmers around Kilimanjaro and analyzed how they decided what to plant in varying environmental conditions. Gould understood the patterns of land-use and the choices made by farmers are the result of decisions made either individually or collectively and that it might be useful to try to model those decisions in a game theoretical framework. In Gould's models the environment is one player and the farmer is another. Each of the players is faced with a number of different strategies the solution of which is the game's equilibrium. Using simple matrix games he was able to construct cartographic representations of various equilibrium alternatives that could be compared to what was in the fields.In his early research, Gould studied a group of African farmers around Kilimanjaro using decision theory to analyze how they decided what to plant in varying environmental conditions. Gould understood the patterns of land-use and the choices made by farmers are the result of decisions made either individually or collectively and that it might be useful to try to model those decisions in a game theoretical framework. In Gould's models the environment is one player and the farmer is another. Each of the players is faced with a number of different strategies the solution of which is the game's equilibrium.

Importantly, Gould realized that the game theory of the time was still algorithmically primitive and that his results determined neither how the farmers actually behaved nor how they should have behaved in an absolute sense, but rather how they should behave if they want to achieve particular results. In strategic games, such as the one Gould proposed in his papers, Nash equilibria are a set of actions amongst the payers that lead to a steady state. It is a position in the game in which each player holds the correct expectation about the other player and behaves and acts rationally according to his choices. Gould uses the simple graphical solutions to the matrix games he creates which I found so attractive early on in Harold Kuhn’s lectures. For more on Kuhn and John Nash watch the video of a recent seminar they gave together at Princeton, here.

The concept of equilibrium is not so straightforward here as one might think, and it can be interpreted in several ways. For example, when we say that a physical system is in equilibrium we might mean that it is in a stable state, one in which all the causal forces internal to the system are in balance. This is the traditional economic meaning of equilibria. The variables are dynamic however, and the balance between them that makes up the equilibrium can be thought of as networks of mutually constraining relations. Equilibria can then be considered as endogenously stable states of the model. Some scholars however, interpret game theoretic equilibria as being explanatory of the process of strategic reasoning alone. For them a solution must be an outcome that a rational agent would predict using the mechanisms of rational computation alone. The meaning of equilibrium states is still a matter of discussion in the literature of game theory and has interesting philosophical implications to how we view and interpret what the models tell us outside of their mathematical formalism. (For more on the interpretation of game theoretical results see Ariel Rubinstein's seminal paper Comments on the Interpretation of Game Theory or thePhilosophy of Game Theory by Grune-Yanoff.

The current models I am working with are of course much more complex than anything Gould could have considered, as he lacked both the mathematics and the computing power. New techniques like quantal response functions, which allow us to look at probable actions, are much more powerful and yield much more interesting results. They were first introduced by McKelvey and Palfrey in the late 1990s and considered mathematically for the possibility that the players will make mistakes and therefore they give more realistic results than anything Gould imagined, at least we hope they do.

The power of models in historical geography is that you can look at many different scenarios and compare them with the little actual historical data you have. I would never assert that what I am doing actually gives me any definitive answers on what decisions Roman farmers made or how they planted, rather they show me what possibilities there were and how to rank them. Most importantly however, they greatly inform my thinking about the Roman economy in its most empirical form, and since I do not have the disciplinary constraints on my ideas that an economic historian might, I can push the limits of the models for purely theoretical and curiosity reasons.My hope is that these methods will yield an 'experimental' historical geography, an acceptance of simulation as a method in historical studies. These simulations have the potential to shed light on the decision alternatives that face farmers and estate owners acting within primitive or developing economies. They give us a glimpse into how historically farmers interacted with their environment on a mainly cognitive level, allowing us to consider the choices they made and how their decisions affected the landscape around them. This to me, and to other geographers before me, like Gould and Harvey, is certainly a central geographical question.

For those interested I am using a software package that can calculate the Nash equilibria for games with large numbers of players, or in this case environmental variables called GAMBIT.http://www.gambit-project.org/doc/index.htmlIt is an open source program and you can have a great deal of fun experimenting with variables and how they change the equilibrium outcome.

My article in Alpinist Magazine 41....called Vericality: the other blank on the map....

Exploring the Ruins of Tikal...

when not climbing in the Alps, mountain biking through some jungle or looking for Roman ruins in North Africa, is a Specialist in Modern Cartography and Geographic Information Science in the Geography and Map Division of the Library of Congress. A Fellow of the Royal Geographical Society he is the recent recipient of a J. L Heiberg Research and Exploration Fellowship for his work on the physical remains of Roman Centuriation and in 2010 was awarded a J.S. Best Fellowship from the American Geographical Society.
While at the Library of Congress, much of his research has concentrated on the use of computer modeling in the analysis of Roman, Medieval and Renaissance maps.
He is also a lecturer on the history of cartography and and Geographic Information Sciences in the Graduate School for Advanced Studies at the Johns Hopkins University.

Kant in the Wilderness: Thoreau's Geographic Turn

.....click on Henry to read my new article just published by the Thoreau Society

History of GIS...my new article in ArcNews....

...click on above to read.....

Exploring the Mer de Glace...

...and its melting....

In the Land of Melting Glaciers...

...working on medieval land charters and using them to reconstruct early land ownership law....

David Kendall's article "Recovery of Structure From Fragmentary Information" first showed how to use statistical methods to reconstruct geographical layouts and maps from chartularies and medieval deeds....

Kendall's early explorations...

...making maps from fragmentary data...

Kendall working on statistical map reconstuction.....

...I try in my reconstructions to follow in his footsteps....

Recent Appearance on C-Span American History Channel

Click to view video....

Deep in the gorges of Provence....

Eric Shipton surveying in the Karakorum....

...if I could only write about it all like he did....

Recent and forthcoming publications....

...another new article...just published in the Portolan....

In Nietzsche's Shadow: Searching for the Remains of Roman Cartography in Southern France

My newest book just released....

..click on the cover to link to the New York Times Review above....

Around the world...some images from recent travels and mapping projects....

Searching for and mapping the megaliths on the island of Bornholm....

" 'Amateur' field geographers can speak with authority about the clarifying effects on the mind of direct physical danger in the real world and there exists a terrible antagonism between field geographers and armchair academics. Not only do those in their armchairs think and write junk, obfuscation, obscurantism, and endlessly convoluted self-referral to their literature in windowless libraries, they do not care about the human condition.”

--William Bunge, Geography is a Field SubjectArea, 1983

It is Bunge’s words that I remember when I am out in the field in places like Tunisia and Algeria.

Exploring theTomb of Cleopatra II, near Tipasa, Algeria

Summer, 2011

...the tomb as it looked in 1926....

...entrance to the tomb....

...the underground aqueducts and sewers of Tipasa, Algeria...

...nothing but questions... trying to map their extent....Summer 2011...

Heidegger Climbing in the Mountains near Davos....

For Peter Gould.... who showed me there was something in Heidegger that geographers could use...

"Indeed space is still one of the things that is constitutive for the world... ---Martin Heidegger

I had read text after text of Martin Heidegger under the tutelage off one of the finest scholars it has ever been my privilege to meet. The philosopher Joseph Kockelmans always worked with parallel German and English texts, and with the Greek text too if it were appropriate, and slowlyopened up the thinking that lay behind Heidegger's tenacious quest. If anyone had told me we would spend sixteen weeks on the forty-five pages of Heidegger's "Anaximander Fragment," I would have laughed, but at the age of forty seven, I began to read properly.---Peter Gould, Becoming a Geographer

...a must read for all geographers and cartographers....

Wittgenstein & Cartographic Theory

...slightly modified [added to] selections from the Tractatus...

We picture facts to ourselves
A picture [map] presents a situation in logical space, the existence and non-existence of states of affairs.
A picture [map] is a model of reality.
A picture [map] is a fact.
A pictorial [cartographic] form is the possibility that things are related to one another in the same way as elements of a picture [cartographic objects].That is how a picture [map] is attached to reality; it reaches right out to it. It is laid against reality like a measure.
Every picture [map] is at the same time a logical[mathematical][spatial] one. Not every picture is however a spatial one. [every map is].It is as impossible to represent in language anything that contradicts logic as it is impossible to represent by its coordinates a figure that contradicts the laws of space [projection], or to give coordinates of a point that does not exist. [void]

...Wittgenstein's grave at Acsension Parish Burial Ground, Cambridge....notice the little ladder that recalls the last few statements of the Tractatus....

...my theory of cartographic history...

The actual maps that exist are but a tiny subset of the theoretical maps that could exist. These real maps are products of a very small number of trajectories through cartographic space...each with its own unique place in this mathematical construction. Every real map is surrounded by a tiny cluster of real or unreal neighbors who are its ancestors and descendents...

...this notion takes its starting point from David Lewis' "On the Plurality of Worlds".

...think of maps as random graphs.....

...both of these books by Tim Robinson are must reads for any cartographer....or historian of maps.

....Robinson's reflections on cartography and the effect of landscape on abstract thought are among the most insightful in all of cartographic literature.....

Gian-Carlo Rota.....

His MIT lectures changed my way of thinking about mathematics....

His "End of Objectivity" lectures were groundbreaking and are currently not easy to come by.....

Galois Lattices....

....maps are represented using attribute tables and the connections between them are calculated using programs like Galacia....click on lattcies for more on Galacia......more details coming in a future full post...

Garrett Birkhoff....

...his Harvard Lectures on Lattice Theory...the class known as MATH 252...developed some of the most beautiful and applicable algebraic structures to the study of stemma and recensions.

The mathematics of the medieval Portolan chart

Click for the Washington Post Story on my modeling of Portolan Charts

Mathematical models of the 1507 Waldseemuller Map

...click to read the Washington Post article on my research....

This blog is featured at the National Library of Scotland's Georeferencing page

Waldseemuller and me....

Click on image for the video "Bridging the Gap Between Physical and Digital Preservation" about the encasing and digitization of the Waldseemuller map...

My Recent Books...

The Naming of America: Martin Waldseemuller's 1507 World Map and the Cosmographiae Introductio

...this is my translation and commentary on Waldseemuller's Cosmographiae Introductio....click on image to order from Amazon...

Reviews of The Naming of America

Imago Mundi:

....Hessler’s nuanced translation brings to life this dynamic period of cartographic history and the theories used by these early sixteenth-century cosmographers.His close attention to the Latin and his extensive notes reflect a level of serious scholarship that should place this book on the reading list for all graduate seminars focused on understanding the production of early modern knowledge. The presentation, graphic aesthetic and accessibility of the text will make this a favourite for general readers, as well, and should be on the wish list of everyone interested in early American history and cartography.

California Literary Review:

...It lurks in the background of our childhood imagination, now and again roaring back in adulthood to remind us of possibilities. A map of the world, that fixture in elementary classrooms, has always been a book masquerading as a flat piece of paper. Like layers of the earth for geologists, maps offer a glinting sample of the past. And when it comes to the Waldseemüller map, the Universalis Cosmographia that forms the subject of The Naming of America by John W. Hessler, there are earth-shattering discoveries to be found. Let it be said, up front, that The Naming of America is not a popular work in the vein of Doris Kearns Goodwin or Stephen Ambrose. Hessler’s is a scholarly affair, impeccably printed, where the footnotes are as long as the text, and controversies are discussed with dry impartiality....

Warntz and Tobler...two mathematical pioneers....

William Warntz on topological and cartographic surfaces....

We now look upon maps not only as stores for spatially ordered information, but also as a means for the graphical solution of certain spatial problems for which the mathematics proves to be intractable, and to produce the necessary spatial transformations for hypothesis testing....The modern geographer concieves of spatial structures and spatial processes as applying not only to such things as landforms....but also to social, economic, and cultural phenomena portraying not only conventional densities but other things such as field quantity potentials, probabilites, refractions etc etc. Always these conceptual patterns may be regarded as overlying the surface of the real earth and the geometrical and topological characteristics of these patterns, as tranformed mathematically or graphically, thus describe aspects of the geography of the real world...

---Spatial Order, Harvard Papers in Theoretical Geography 1, 1967

We recognize yet another role for maps. In the solution of certain problems for which the mathematics, however elegantly stated, is intractable, graphical solutions are possible. This is especially true with regard to "existence theorems". There are many cases in which the graphical solution to a spatial problem turns out to be a map in the full geographical sense of the term, "map." Thus a map is a solution to the problem.

The basic ideas of the theory of convex sets are naturally and easily appreciated when examined in a geographical context, rather than in a non-spatial one...in this respect geography is truly elegant.

Maps showing regional classification can be regarded as logic diagrams. Mapping of sets is a general mathematical concept. Geographical mapping is merely a special case of this.

Tobler's computer program, called Bi-dimensional Regression marks the real beginning of the mathematical analysis of historic maps. The paper was published in 1977 along with a Fortran program which performed an 'empirical transformation regressing an independent plane configuration against a similar configuration.' The program itself constructs this geometric transformation by estimating two sets of bivariate points on a square lattice that have been interpolated from the originally irregularly arranged original observations. This is very much analogous to modern landmark morphometrics using thin-plate splines and various other methods. The curvilinear regression coefficients are represented by Tobler as a spatially varying, but coordinate invariant, second-order tensor field...

Tobler's paper is geometrically insightful and, although being from the 1970's, still contains ideas that are yet to be fully explored in the study of the history of cartography.... Two-dimensional asymmetric tensor analysis for example... his work inspires everything I do....

By far the best theoretical work on the methodology of the history of cartography.

What the historian of cartography should be concerned with is a systematic study of the factors affecting error, and seek to establish their cause and variability and the statistical parameters by which error is characterized.....